A precision calculation of the effective number of cosmological neutrinos

G. Mangano; G. Miele; M. Peloso; S. Pastor

REVIEW

A precision calculation of the effective number of cosmological neutrinos

G. Mangano, G. Miele, S. Pastor, M. Peloso

Astrophysics · Submitted 21 Nov 2001 · 32 references

Pith's verdict: A careful, parameter-free joint calculation that lands on N_eff = 3.0395; the useful structural point is that QED and transport corrections overlap, though the paper never runs the ablation that would nail it.

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CitationG. Mangano, G. Miele, M. Peloso, S. Pastor. "A precision calculation of the effective number of cosmological neutrinos." arXiv:astro-ph/0111408 (2001). https://pith.science/paper/2KKDJRQN

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Editor's note

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This is the paper that gives you N_eff = 3.0395 for three massless neutrinos, and it's worth having on the shelf. The ingredients are all published — the transport code is the authors' own from Esposito et al. [8], the finite-temperature QED mass shifts are Heckler [3] — but running them in one coupled system is the new bit, and the result is a genuinely new number. The no-QED limit reproduces both their prior 3.0345 and Dolgov-Hansen-Semikoz's 3.0340 ± 0.0003, which is a real cross-check by a different numerical method. Table 3 lays out every literature value side by side, including the Gnedin & Gnedin 3.022 they disagree with, and the extra-relics and massive-neutrino sections are sensible extensions rather than padding. The equations are written out explicitly (Eqs. 3–22), so the calculation is checkable by anyone willing to reimplement it. The soft spot is the non-additivity claim. The paper says the QED correction on top of transport is 0.005, 'a factor 2 less' than the 0.011 from refs [3,12], and attributes the difference to the G2 term in the denominator of Eq. (17). The equation structure does support the qualitative point — G2 sits alongside the neutrino df/dx terms, so the two effects can't be cleanly separated. But the 0.011 baseline comes from instantaneous-decoupling calculations, not from their own code. The clean test is to run their code with transport off and QED on; if that gives ~0.011 the claim stands, if it gives ~0.005 the 'factor 2' is a comparison artifact. They don't do it. I'd call this a fixable gap, not a flaw, but it's the one thing I'd ask for. Two smaller things. The headline carries four digits with no error bar, while the paper itself says n=3 truncation gives ~1% accuracy in the distortions — that should have been propagated. And 'perfect agreement' with CMBFAST's 3.04 is a rounded default, not validation; the agreement is real but the phrasing oversells it. Who should read it: anyone doing CMB parameter extraction or BBN who needs the theoretical N_eff and its provenance. It deserves a serious referee — the calculation is standard, the code is theirs and previously published, and the numbers are checkable.

Referee report

Major objections (3)

[Section 3, Table 3] The non-additivity claim compares the joint result (3.0395) to a QED-only shift of 0.011 taken from refs. [3,12], which treat QED in the instantaneous-decoupling limit. The comparison therefore mixes two approximations: the joint calculation has transport on, while the QED-only baseline has transport off. To establish non-additivity within a single framework, the authors should perform the ablation with their own code (transport terms dfν/dx set to zero in Eq. 17, G1 and G2 retained) and report the resulting QED-only ΔN_eff. If that ablation gives approximately 0.011, the claim is supported; if it gives approximately 0.005, the joint result is simply additive and the '10% smaller than the naive sum' statement is an artifact of comparing to a literature value computed under different assumptions. At minimum, the language should be softened to reflect that the comparison spans two approximations.
[Section 3, Eq. (17)] The functions G1 and G2 in Eq. (17) are the novel technical ingredient and the stated mechanism for non-additivity, yet they are given without derivation. A short derivation, or a numerical check that G1 and G2 vanish as α → 0 and reproduce the known QED equation-of-state correction in the appropriate limit, would make the result reproducible and guard against an error that would be indistinguishable from genuine interplay. As written, a reader cannot verify the central structural claim from the manuscript alone.
[Section 2, Eq. (5) and Table 2] The truncation of the neutrino distribution expansion at n = 3 is stated to give about 1% accuracy in the neutrino distortions, but the propagation of that truncation error to N_eff is not quantified anywhere, while the headline 3.0395 is quoted to four digits. The agreement with refs. [6,7] (3.0340 ± 0.0003) in the no-QED limit mitigates this concern, but an explicit error estimate, for example an n = 4 or n = 5 comparison, would strengthen the claim and make the four-digit quotation defensible.

Minor objections (9)

[Section 2 title] The section title reads 'Corrections from to the non instantaneous decoupling'; the stray 'from' should be removed.
[Section 3] The phrase 'the r.h.s. of Eq.s (3) and (7)' should read 'Eqs. (3) and (7)'.
[Section 4] The sentence 'it is seems more and more clear' contains a grammatical error and should read 'it seems more and more clear'.
[Section 4] The phrase 'the presence of a finite mass modify of course' should read 'modifies'.
[Section 4] The phrase 'active neutrino are fully relativistic' should read 'active neutrinos are fully relativistic'.
[Section 5] The phrase 'Despite of the smallness' should read 'Despite the smallness'.
[Section 5] The phrase 'the main result of our analysis, which have been carried out' should read 'which has been carried out'.
[Section 3, Eq. (12)] The momentum-dependent term in the electron mass correction is dropped with the claim that it contributes less than 10%; a brief justification or a reference supporting that estimate would help the reader.
[Figure 1] The figure would be clearer if the temperature correspondence to the x-axis were marked directly on the plot rather than only described in the caption and text.

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The paper

The paper claims

Solving neutrino decoupling and finite-temperature QED plasma corrections in one coupled system gives an effective neutrino number $N_eff = 3.0395$ for three massless active neutrinos, confirming the value CMB codes already assume and under

The reading

The paper asks a narrow but consequential question: how much energy does the cosmic neutrino background actually carry today? In the standard instantaneous-decoupling picture the answer is exactly three neutrino species' worth, but two small effects push it upward: neutrinos keep interacting weakly while electrons and positrons annihilate, and the electromagnetic plasma's own equation of state is modified by finite-temperature QED corrections. Earlier work treated these two effects separately and added them, arriving at about 3.045. The authors solve the neutrino Boltzmann equations with both effects switched on simultaneously, and find that the corrections overlap rather than stack. The result, $N_eff = 3.0395$, sits within a thousandth of the default value hard-coded into the widely used CMBFAST code, so the number cosmologists have been plugging into microwave-background fits is, on this calculation, the right one.

What carries the argument

The machinery is a coupled system: Boltzmann equations for the electron-neutrino and the combined muon/tau distributions, expanded in orthonormal polynomials over the Fermi-Dirac weight and truncated at four terms, solved together with a continuity equation for the total plasma energy density and pressure. The load-bearing object is that continuity equation, rewritten as an evolution equation for $z = T R$. Finite-temperature QED enters through a thermally shifted electron mass and a photon thermal mass, which lower the plasma energy density and hence the entropy released into photons. Because these corrections appear as the functions $G_1$ and $G_2$ inside the same equation that carries the neutrino distortion terms, the two effects are entangled rather than separable, and that entanglement is what produces the 0.005 rather than 0.011 shift.

Extended reading notes

Core claim

The central claim is that a single, self-consistent Boltzmann calculation of neutrino decoupling, with the photon and $e^+e^-$ plasma described by its finite-temperature QED equation of state, yields $N_eff = 3.0395$ for three massless active neutrinos, with $delta z/z_0 = -1.841 times 10^{-3}$, $delta rho_{nu_e}/rho_nu^0 = 0.935%$ and $delta rho_{nu_x}/rho_nu^0 = 0.390%$. The authors stress that the two corrections cannot be added linearly: the QED terms enter the very continuity equation that governs the photon-to-neutrino temperature ratio, so the neutrino non-thermal tails and the plasma corrections act on each other. The combined shift is 0.005 in $N_eff$, roughly half of what a naive sum of the separate effects would predict, and the naive value 3.045 is therefore excluded at the accuracy the authors claim. They also extend the result to extra relativistic relics, finding $N_eff = 3.0395 + (1-beta)(z_0/z_fin)^4 N_X$ with $beta = 0.0014$ for $0 < N_X < 1$, and show that for a neutrino mass around 1 eV the fractional energy excess from incomplete decoupling fades as the temperature drops below the mass.

Load-bearing premise

The neutrino distribution functions are expanded in only four orthogonal polynomials, a truncation the authors themselves say is accurate to about 1 percent in the distortions, and no error budget is propagated from that truncation to the four-digit value of $N_eff$.

Editorial extensions

If this is right

Cosmological fits that assume $N_eff = 3.04$ for three massless neutrinos are using a value correct to better than a thousandth, so no re-analysis of existing microwave-background data is forced by this recalculation.
Any future measurement claiming to resolve $Delta N_eff$ at the 0.005 level must account for the fact that the standard-model baseline is 3.0395, not 3.0, and that the two corrections to it are not additive.
Extra relativistic species present at neutrino decoupling contribute with a small suppression factor, $(1-beta) = 0.9986$, on top of the usual dilution by the photon-to-neutrino temperature ratio.
For neutrino masses near 1 eV, the non-thermal energy excess from incomplete decoupling shrinks as the universe cools and freezes into a constant offset in number density rather than energy density.
Big Bang nucleosynthesis is unaffected at any observable level: the change in the helium-4 mass fraction is of order $10^{-4}$, below both theoretical and experimental uncertainties.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's own cross-check against an independent grid-based solver is where the real uncertainty lies: the two calculations agree on the no-QED baseline to about $3 times 10^{-4}$ in $N_eff$, which is the same order as the QED shift being claimed, so the headline precision rests on a comparison that is only marginally tighter than the effect.
If a future calculation with a finer momentum grid and a higher polynomial truncation moves $N_eff$ by more than about 0.001, the interesting physics would not be the shift itself but the demonstration that four-term polynomial expansions of a distorted Fermi-Dirac tail are inadequate at this precision.
The non-additivity argument generalises: any early-universe correction that modifies the plasma equation of state will entangle with the neutrino transport equations in the same way, so separate-and-add estimates of small relativistic corrections are systematically unreliable at the $10^{-3}$ level.
Because the distortion is momentum-dependent, with higher-momentum neutrinos heated more, a neutrino mass ordering or a non-standard momentum distribution would change the size of the effect in a way the paper's massless-plus-1-eV treatment does not explore.

Original abstract

The neutrino energy density of the Universe can be conveniently parametrized in terms of the so-called effective number of neutrinos, N_nu^eff. This parameter enters in several cosmological observables. In particular it is an important input in those numerical codes, like CMBFAST, which are used to study the Cosmic Microwave Background anisotropy spectrum. By studying the neutrino decoupling with Boltzmann equations, one can show that this quantity differs from the number of massless neutrino species for an additional contribution due to a partial heating of neutrinos during the electron-positron annihilations, leading to non thermal features in their final distributions. In this paper we review the different results obtained in the literature and perform a new analysis which takes into account, in a fully consistent way, the QED corrections at finite temperature to the photon and e^+- plasma equation of state. The value found for three massless active neutrinos is N_nu^eff=3.0395, in perfect agreement with the recommended value used in CMBFAST, N_nu^eff=3.04. We also discuss the case of additional relativistic relics and massive active neutrinos.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,726 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The neutrino energy density of the Universe can be conveniently parametrized in terms of the so-called effective number of neutrinos, N_nu^eff. This parameter enters in several cosmological observables. In particular it is an important input in those numerical codes, like CMBFAST, which are used to study the Cosmic Microwave Background anisotropy spectrum. By studying the neutrino decoupling with Boltzmann equations, one can show that this quantity differs from the number of massless neutrino species for an additional contribution due to a partial heating of neutrinos during the electron-positron annihilations, leading to non thermal features in their final distributions. In this paper we review the different results obtained in the literature and perform a new analysis which takes into account, in a fully consistent way, the QED corrections at finite temperature to the photon and e^+- plasma equation of state. The value found for three massless active neutrinos is N_nu^eff=3.0395, in perfect agreement with the recommended value used in CMBFAST, N_nu^eff=3.04. We also discuss the case of additional relativistic relics and massive active neutrinos.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The neutrino energy density of the Universe can be conveniently parametrized in terms of the so-called effective number of neutrinos, N_nu^eff.	background 1.00	0.05	0.03	asserted 0.99
s1	This parameter enters in several cosmological observables.	background 1.00	0.04	0.02	asserted 1.00
s2	In particular it is an important input in those numerical codes, like CMBFAST, which are used to study the Cosmic Microwave Background anisotropy spectrum.	background 1.00	0.05	0.02	asserted 1.00
s3	By studying the neutrino decoupling with Boltzmann equations, one can show that this quantity differs from the number of massless neutrino species for an additional contribution due to a partial heating of neutrinos during the electron-positron annihilations, leading to non thermal features in their final distributions.	background 0.97	0.16	0.05	asserted 1.00
s4	In this paper we review the different results obtained in the literature and perform a new analysis which takes into account, in a fully consistent way, the QED corrections at finite temperature to the photon and e^+- plasma equation of state.	contribution 0.90	0.86	0.05	asserted 0.88
s5	The value found for three massless active neutrinos is N_nu^eff=3.0395, in perfect agreement with the recommended value used in CMBFAST, N_nu^eff=3.04.	result 1.00	0.95	0.81	asserted 1.00
s6	We also discuss the case of additional relativistic relics and massive active neutrinos.	contribution 0.98	0.62	0.03	unclear 0.50

What kind of paper the classifier reports this is: theoretical result 0.84 · evidence from simulation or computation 0.87.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Evolution of δρνα (mα)/ρ0 να (mα) , for a neutrino mass mα = 1 eV (see text for further details). of Section 3, it is easy to see that δρνα (m)/ρ0 να (m) is given by δρνα (mα) ρ 0 να (mα) = 10−4 R ∞ 0 dy y2 r y 2 + mα me 2 x 2 (ey + 1)−1 P3 i=0 c α i y i R ∞ 0 dy y2 r y 2 + mα me 2 x 2 (ey + 1)−1 . (29) In

Figures reproduced from arXiv:astro-ph/0111408 by the authors.

References

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A dark SU(2) sector with two Higgs doublets and a neutrino portal could generate the baryon asymmetry via a first-order phase transition, predicting ΔN_eff = 0.09-0.13 and exotic Higgs/Z decays, though the asymmetry y...
New physics in light of the $H_0$ tension: an alternative view
astro-ph.CO 2019-07 conditional novelty 5.0 of 10

Fixing the dark energy equation of state near -1.3, or the relativistic species count near 3.95, makes the CMB plus BAO plus supernova estimate of the Hubble constant coincide with the local ladder value, at a steep B...
Neutrino decoupling beyond the Standard Model: CMB constraints on the Dark Matter mass with a fast and precise $N_{\rm eff}$ evaluation
hep-ph 2018-12 conditional novelty 5.0 of 10

A simplified two-equation treatment of neutrino decoupling reproduces the Standard Model N_eff to 0.01 and yields CMB bounds excluding electrophilic or neutrinophilic thermal dark matter below 3 MeV.
Light dark states with electromagnetic form factors
hep-ph 2018-11 accept novelty 5.0 of 10

Combining accelerator searches, precision measurements, and cosmology rules out dark matter that couples to photons through magnetic dipole, electric dipole, anapole, or charge-radius interactions, assuming a standard...
The Simons Observatory: Science goals and forecasts
astro-ph.CO 2018-08 conditional novelty 5.0 of 10

The paper specifies the Simons Observatory's telescopes, frequency bands, noise model and foreground treatment, and forecasts uncertainties on the tensor-to-scalar ratio, relativistic species, neutrino mass, lensing, ...
Severely Constraining Dark Matter Interpretations of the 21-cm Anomaly
hep-ph 2018-03 conditional novelty 5.0 of 10

After applying CMB, BBN, supernova, and lab constraints, only a millicharged dark matter component making up 0.3 to 2 percent of the dark matter, with mass 10 to 80 MeV and charge 10^-6 to 10^-4, can still explain the...
Searching for dark matter - dark energy interactions: going beyond the conformal case
astro-ph.CO 2017-09 conditional novelty 5.0 of 10

Joint cosmological fits find no significant conformal or disformal coupling between dark matter and dark energy, with 95% upper limits near alpha < 0.03 and D_M < 0.3 meV^-1 once cluster abundances are included.
Relic neutrino decoupling with flavour oscillations revisited
hep-ph 2016-06 accept novelty 5.0 of 10

Full off-diagonal collision integrals in relic neutrino decoupling alter the frozen neutrino spectra but leave the effective number of neutrinos at N_eff = 3.045, independent of the neutrino mass ordering.
Neutrino energy transport in weak decoupling and big bang nucleosynthesis
astro-ph.CO 2015-12 conditional novelty 5.0 of 10

Self-consistently coupling Boltzmann neutrino transport to the big bang nucleosynthesis network shifts primordial deuterium by roughly 0.4 percent and helium by about 0.05 percent relative to no-transport calculations.
Dark Radiation constraints on minicharged particles in models with a hidden photon
hep-ph 2013-11 conditional novelty 5.0 of 10

A full numerical treatment of minicharged-particle thermalization excludes hidden-sector fermions lighter than roughly a GeV unless their effective charge is below about 10^-9 to 10^-7.
A Lower Bound on the Mass of Cold Thermal Dark Matter from Planck
hep-ph 2013-03 accept novelty 5.0 of 10

Cold thermal dark matter that remained in equilibrium with neutrinos, or with electrons and photons, after neutrino decoupling must be heavier than roughly 1 to 10 MeV, because its entropy transfer shifts the neutrino...
Increasing Neff with particles in thermal equilibrium with neutrinos
astro-ph.CO 2012-07 conditional novelty 5.0 of 10

A sub-10 MeV particle in thermal equilibrium with neutrinos until it goes non-relativistic raises Neff above three and predicts Neff(CMB) > Neff(BBN), a pattern current data cannot yet exclude.
Joint probes of dark matter annihilation from neutrino detectors and CMB targets
hep-ph 2026-06 unverdicted novelty 4.0 of 10

Joint analysis of neutrino detectors and CMB observables can constrain dark matter annihilation into neutrinos for MeV-GeV masses.
Probing Dark Sector Particles Coupling to Neutrinos with Double Beta Decay
hep-ph 2025-11 unverdicted novelty 4.0 of 10

Double beta decay experiments can constrain couplings of sub-MeV Majoron-like scalars to neutrinos at |a_ν| ≈ 2×10^{-6} through on- and off-shell production effects on the electron spectrum.
Inflationary interpretation of the stochastic gravitational wave background signal detected by pulsar timing array experiments
astro-ph.CO 2023-06 conditional novelty 4.0 of 10

An inflationary origin for the nanohertz gravitational wave background reported by pulsar timing arrays would require a tensor spectral index near 1.8 and reheating below about 10 GeV, which makes it hard to sustain.
A precision calculation of relic neutrino decoupling
hep-ph 2020-05 conditional novelty 4.0 of 10

Solving the full neutrino decoupling kinetic equations with three-flavor oscillations and next-to-leading QED corrections gives N_eff = 3.044, with an estimated error of 0.0005.
Alleviating the $H_0$ and $\sigma_8$ anomalies with a decaying dark matter model
astro-ph.CO 2019-02 conditional novelty 4.0 of 10

A one-parameter decaying dark matter model reduces both the Hubble and S8 tensions when fitted to Planck plus a local H0 prior, but the improvement is not statistically significant and weakens when supernova and BAO d...
Non-Thermal Dark Matter Mimicking An Additional Neutrino Species In The Early Universe
astro-ph.CO 2011-11 conditional novelty 4.0 of 10

A sub-percent fraction of dark matter produced in late decays can reproduce the expansion history of one extra neutrino species while evading light-element constraints.
PArthENoPE: Public Algorithm Evaluating the Nucleosynthesis of Primordial Elements
astro-ph 2007-05 conditional novelty 4.0 of 10

The paper releases and documents PArthENoPE, a Fortran code that solves the Big Bang nucleosynthesis reaction network and predicts light-element abundances for standard and non-standard cosmological inputs.
A search for sterile neutrinos with the latest cosmological observations
astro-ph.CO 2017-03 conditional novelty 3.0 of 10

Combining Planck 2015 with BAO, H0, SZ cluster counts, lensing, and cosmic shear gives Neff = 3.29 +0.11 -0.17, a 1.44-sigma hint of dark radiation, and a 95% upper limit of 0.24 eV on a massive sterile neutrino.
Axion Cosmology
astro-ph.CO 2015-10 accept novelty 2.0 of 10

Axions can act as dark matter, dark energy, or dark radiation depending on mass, and existing cosmological data already exclude a single ultralight axion species below about 10^-24 eV from composing all of the dark matter.
Non Thermal Features in the Cosmic Neutrino Background
astro-ph 2006-03 unverdicted novelty 1.0 of 10

A review of current cosmological bounds on distortions of the cosmic neutrino background, concluding that present data only weakly constrain non-thermal features and future experiments will tighten some, but not all, ...
Massive neutrinos and cosmology
astro-ph 2006-03 accept novelty 1.0 of 10

A detailed review of how cosmological observations bound the sum of neutrino masses through the free-streaming suppression of small-scale structure, and of future sensitivity forecasts.

How later papers used it

112 later papers cite it

Evidence: substantial. 108 of the citing papers share no author with it, and 49 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 184 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

6 later papers by other authors are listed below: 2 used its method or tool, 3 used its data or values and 1 compared its own results with it.

Uses its method or tool 2 of 10 papers with this label

Minimized the Gibbs free energy to find this paper's expression for the energy density.

Insights into neutrino decoupling gleaned from considerations of the role of electron mass (2017) · 0.90 · paraphrased: the licence does not allow quoting
Following reference [ref] , we introduce the comoving quantities $x≡m_e a$, $y≡a p$, and $z ≡T a$, and rewrite the continuity equation [ref] as a differential equation for the quantity $z$ (corresponding to the photon-to-neutrino temperature ratio): [equation] where $J(x/z)$ and $Y(x/z)$ describe the ideal-gas behaviour of the QED plasma, while the $G_1,2(τ)$ account for deviations of the QED plasma from an idea gas.

Towards a precision calculation of $N_{\rm eff}$ in the Standard Model III: Improved estimate of NLO contributions to the collision integral (2024) · Solving the neutrino Boltzmann equations and the continuity equation · 0.99

Uses its data or values 3 of 41 papers with this label

Compared the 2015 Planck constraint on N_eff with this paper's SM prediction of 3.046.

Non-Abelian dark matter and dark radiation (2015) · 0.94 · paraphrased: the licence does not allow quoting
Used this paper's value of 3.046 for the effective number of neutrino species for 3 types of active neutrinos.

Relic Neutrino Degeneracies and Their Impact on Cosmological Parameters (2020) · 0.93 · paraphrased: the licence does not allow quoting
Used this paper's value of 3.046 for ordinary SM neutrinos in its radiation density.

Phases of New Physics in the CMB (2015) · 0.88 · paraphrased: the licence does not allow quoting

Compares with it 1 of 4 papers with this label

Checked that its own equation and the corresponding expression in this paper are numerically identical to within machine precision.

Towards a precision calculation of the effective number of neutrinos $N_{\rm eff}$ in the Standard Model I: The QED equation of state (2019) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

102 arXiv papers cite this in 171 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 94 sentences without a confident label

Uses its method or tool 10 papers

Currently, the most widely used computational procedure to accounts for these equation of state effects in the context of $N_eff$ is that documented in [ref] , which follows the same heuristic thermal mass arguments of [ref] to $O(e^2)$, where $e$ is the elementary electric charge.

Towards a precision calculation of the effective number of neutrinos $N_{\rm eff}$ in the Standard Model I: The QED equation of state · Introduction · 0.83
We use the finite temperature corrections at leading order in $α$ from [ref] and [ref] , and proceed as [ref] to obtain the temperature evolution of the electromagnetic plasma including finite temperature corrections: [equation] where $P_int$ reads [ref] : [equation] and the leading order QED finite temperature correction to the masses are [ref] : [equation] where $α$ is the fine structure constant, and for $δm_e^2(T)$ we have neglected a momentum dependent contribution that amounts to less than 10% for an average electron [ref] .

Neutrino decoupling beyond the Standard Model: CMB constraints on the Dark Matter mass with a fast and precise $N_{\rm eff}$ evaluation · Finite temperature QED corrections · 0.97
The differential equations which the code solves are the following, written in terms of the comoving coordinates $x≡m_e a$, $y≡p a$ and $z≡T_γ a$ [ref] :

Thermalisation of sterile neutrinos in the early Universe in the 3+1 scheme with full mixing matrix · Equations · 0.93

Uses its data or values 36 papers

N_ν= 3.046+Δ^ν' T_ν/T_ν^4, where the first term is the contribution from SM neutrinos [ref] and the second term from twin neutrinos, as in Eq. [ref].

A Predictive Mirror Twin Higgs with Small $\mathbf{Z}_2$ Breaking · the helium mass fraction · 0.85
We also show the $ΔN_eff$ constraint[ We take the current limit on $N_eff$ as $2.99^+0.34_-0.33$ at the $95%$ confidence level [ref] while we take the SM value of $N_eff$ by $3.04$ [ref] .

Constraining MeV to 10 GeV majoron by Big Bang Nucleosynthesis · Exclusion · 0.99
The predicted value is $N_eff=3.044$ [ref] , and we define $≡N_eff - 3.044$.] $N_eff$ due to the population of a single species of RH neutrinos at a given time is obtained by opportunely integrating $f^R$ over the phase space: [equation] In Fig. [ref], we show the evolution of $ΔN_eff(T)$ for different values of $μ$ taking the initial temperature to be $T_in=100 GeV$.

Strong cosmological constraints on the neutrino magnetic moment · Cosmological implications · 0.95

Compares with it 4 papers

For instance [ref] is formally different from the one in [ref] , while [ref] matches formally with [ref] , but not with [ref] .

Neutrino decoupling including flavour oscillations and primordial nucleosynthesis · Energy conservation and QED equation of state · 1.00
However, we have checked that, numerically, equation ([ref]) and the corresponding expression in [ref] are identical to within machine precision for a range of $τ$ values.

Towards a precision calculation of the effective number of neutrinos $N_{\rm eff}$ in the Standard Model I: The QED equation of state · Log-independent contribution ($ln$) · 0.98
Ref. [ref] uses the perturbative approach outlined in Ref. [ref] compared to our nonperturbative approach.

Lepton asymmetry, neutrino spectral distortions, and big bang nucleosynthesis · All processes · 0.80

Cites it as background 13 papers

These corrections have been considered in a number of previous works (e.g., [ref] ), wherein the departure of the plasma from an ideal gas due to interactions had been invariably formulated in terms of the acquisition of temperature-dependent masses by the interacting particles.

Towards a precision calculation of the effective number of neutrinos $N_{\rm eff}$ in the Standard Model I: The QED equation of state · Non-ideal gas: Finite-temperature corrections to the QED equation of state · 0.91
A precision calculation of $N_eff^SM$ requires one to solve the process of neutrino decoupling in the early Universe and this has been a subject of intense study for more than 40 years now, see [ref] for references devoted to this problem, see [ref] for reviews and [ref] for a book.

Neff in the Standard Model at NLO is 3.043 · Introduction · 1.00
However, the overlap between neutrino decoupling and $e^±$ annihilations, known as incomplete neutrino decoupling, leads to slightly non thermal neutrino spectra, and to an increased neutrino energy density (both typically of order 1%), which is usually described by an effective number of thermalised neutrinos $$ departing from $3$ [ref] .

Neutrino decoupling including flavour oscillations and primordial nucleosynthesis · Introduction · 0.82

Discussion

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